Question: What is the extraneous solution to these equations? $\dfrac{x^2 - 4}{x + 7} = \dfrac{-17x - 74}{x + 7}$
Explanation: Multiply both sides by $x + 7$ $ \dfrac{x^2 - 4}{x + 7} (x + 7) = \dfrac{-17x - 74}{x + 7} (x + 7)$ $ x^2 - 4 = -17x - 74$ Subtract $-17x - 74$ from both sides: $ x^2 - 4 - (-17x - 74) = -17x - 74 - (-17x - 74)$ $ x^2 - 4 + 17x + 74 = 0$ $ x^2 + 70 + 17x = 0$ Factor the expression: $ (x + 10)(x + 7) = 0$ Therefore $x = -10$ or $x = -7$ At $x = -7$ , the denominator of the original expression is 0. Since the expression is undefined at $x = -7$, it is an extraneous solution.